Regular Tur\'an numbers and some Gan-Loh-Sudakov-type problems

TL;DR

This paper introduces regular Turán numbers, a variation of classical Turán numbers with regular host graphs, and explores their properties, including a supersaturation version of Mantel's theorem and characterizations of graphs based on their regular Turán behavior.

Contribution

It defines regular Turán numbers and analyzes their properties, providing new results such as a supersaturation theorem and characterizations of graphs.

Findings

Supersaturation version of Mantel's theorem for regular graphs of odd order

Characterization of graphs with classical or non-classical regular Turán behavior

Introduction of regular Turán numbers as a new graph extremal concept

Abstract

Motivated by a Gan-Loh-Sudakov-type problem, we introduce the regular Tur\'an numbers, a natural variation on the classical Tur\'an numbers for which the host graph is required to be regular. Among other results, we prove a striking supersaturation version of Mantel's theorem in the case of a regular host graph of odd order. We also characterise the graphs for which the regular Tur\'an numbers behave classically or otherwise.